Continuous Regular Functions
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Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function f:[0,1] → [0,1] is r-regular if there is a Büchi automaton that accepts precisely the set of base r ∈N representations of elements of the graph of f. We show that a continuous r-regular function f is locally affine away from a nowhere dense, Lebesgue null, subset of [0,1]. As a corollary we establish that every differentiable r-regular function is affine. It follows that checking whether an r-regular function is differentiable is in PSPACE. Our proofs relies crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.
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